In fact, the analogy between Iwasawa polynomial and Alexander polynomial via etale homotopy and maximally extended Galois group.
Firstly, we introduce 4-gon region category and signed 4-gon region category. Signed 4-gon region category can represent all the possible Alexander polynomial and all the possible categorical generalization by using n-gons. And more, we will explain the algebraic richness of their degeneration, and, the meaning of complexity using the orderd set of complication functions.
Secondly, we will talk about topological generalization about signed 4-gon region category depend on topological characters via surfaces and cusps.
Thirdly, we will discuss the correspondence between combinatorial explosion by binary gauss phrase and 4-gon region category, and, homotopy theoretic advantage of region category for Iwasawa theory on cyclotomic fields.
Finally, we will show the some possibilities by binary gauss phrase about multiple zeta values in the context of probability theory.
We are using several models of hyperbolic plane (i.e. Poincare disk model, upper-half plane model), development of surface is the most common property between their models, and, several matrix operations depended.
Firstly, we will explain symbolic construction of development of surface, this is a general construction of their models. And more, in the context of invariant functions, we will discuss one theoretical limit of this general construction, and some explanation of this ambiguity related to several topics of topology and geometry.
Secondly, we introduce the 4-gon region category of surface groups and pre-computable category. The 4-gon region category of surface groups is a generalization for their theoretical limit of symbolic construction of development of surface, and pre-computable category is a generalization of "computable", "calculable" and "provable" classes for category theory. Pre-computable category is a theory for all the possible algorithms and all the possible invariant functions, to applying for topological object, we will talk about more generalization of the 4-gon region category of surface groups fused into pre-computable category.
Our main reasons are almost-any closed surfaces have hyperbolic structure, we can study their category with respect to representation of fundamental group about closed surface, besides, their teichmuller space and moduli space, similarly, via all the possible topological types accompanied by their computable classes.
Finally, in the context of homotopy theory, we will discuss between some details of pre-computable category and Brown's representation theorem, especially, purity of invariant functions and a theory for evaluation of pre-computable category.